An Improved Model for High Temperature Inductive Heating

Transcription

1 An Improved Model for High emperature Inductive Heating S. A. Halvorsen * 1 and N. Kleinnecht 1 1 enova AS, Kristiansand, Norway *Corresponding author: sah@tenova.no Abstract: A multiphysics model for industrial induction furnaces has successfully been converted from COMSOL version 3.5a, via versions 4.0a and 4.1, to version 4.2. he model combines computation of magnetic fields, heat transfer and thermal stresses. he inner part of the furnace is described by a few discrete state variables. While the current in the induction coil is input for COMSOL, the model applies an equation for power control, temperature control or a combined power/temperature control. A few case studies have been selected to show differences between the control alternatives. he model can be applied to study operational conditions, thermal stresses, or design details. Compared to version 3.5a the current model has been extended to handle quasi-stationary computations, conductive or non-conductive uid content, far more flexible dynamics, and temperature or combined control. Keywords: multiphysics, inductive heating, control 1. Introduction High temperature reactors heated by electromagnetic induction are used for various purposes lie metal ing, refining, alloying, degassing, etc. Such a reactor consists of a crucible containing a uid, surrounded by suitable insulating materials. An electric coil is located outside the insulation, c.f. Figure 1. ypical industrial furnaces have maximum power input in the range of 50-20,000 W and frequency in the range of 50-1,000 Hz. An alternating electric current in the coil sets up an alternating magnetic field. his field induces electric currents in the uid and/or the crucible to supply the required heating. When a metal is processed, the heat can be delivered directly into the metal, while a nonconductive uid will require sufficient electric conductivity in the crucible. A simulation model has previously been implemented in COMSOL Multiphysics 3.5a. he model is axially symmetric and taes into account electromagnetic induction, temperature distribution, and thermal stresses. he inner part of the reactor is simplified and described by three discrete (lumped) state variables: mass and temperature of the uid, and (average) radiation temperature of the void [1]. he model has been improved and is now converted to COMSOL Multiphysics 4.2. While Matlab scripts and Matlab interface were required to run the ution procedure for the original model, the ver sequence feature introduced in COMSOL 4.0 can now be utilized. Our models are intended for conceptual studies of design and operation of high temperature reactors. Construction details lie tubes in the lid, pouring spout, steel casing, and other aspects not important for the studies are intentionally omitted, or included by coarse descriptions. Further, axial symmetry is assumed. 2. Model he COMSOL model for version 4.2 includes the following application modes: Magnetic Fields (mf) Heat ransfer (ht) ODEs and DAEs (ge) Solid Mechanics (id) Lid Void Liquid Crucible Support Insulation Coil, modeled as a line element Figure 1 Axially symmetric model geometry

2 2.1 Magnetic Fields Maxwell s equations need to be ved to find the electric power induced in the furnace. For an axially symmetric model COMSOL Multiphysics ves for the non-zero angular component of the magnetic vector potential. Equations and other details are described in COMSOL s documentations [2]. he equation form must be set to Frequency domain (not Study controlled or Stationary ). For an industrial furnace the electromagnetic field should not spread outwards. Field guides are therefore located immediately outside the coil. he spread of the field is also limited by a steel casing and possibly also by one or more copper plates. he computational domain for the electromagnetic model must include the coil, and some space outside the coil. It is further limited by the steel casing of the furnace. he field guides can be represented by a perfect magnetic conductor boundary condition, while the steel casing and copper plates are considered as magnetic insulation, c.f. [2]. Axial symmetry is the appropriate boundary condition along the center line. he coil is described by a line element through the center of the turns from the bottom to the top of the coil, c.f. Figure 1. he individual turns might have been represented, which would have made the geometry considerably more complex. Preliminary computations have shown that such details would not significantly improve the accuracy of the computed power distribution. Immediately above and below the electric coil there can be a few turns that are not connected to external current. hese turns are used for additional water cooling. Preliminary calculations showed that induced electric currents in such turns will limit the vertical spread of the magnetic field. In the model they can be appropriately represented by horizontal line elements with a magnetic insulation condition. he surface current density in the angular direction for the line element representing the coil is the appropriate model input parameter. But for the simulations studies the value of this current is not given a priori. Instead, either the total induced power or the temperature in some location can be specified. Hence, an appropriate constraint should be included and the current density in the coil should be adapted to fulfill the constraint. In the previous model it was assumed that the magnetic field lines would be parallel to the field guides outside the coil. Hence, magnetic insulation was applied as a boundary condition also for this region [1]. his idealization can be valid to compute a reasonable approximation for the distribution of the induced heating. But when the boundary is close to the coil, the model will compute high electric boundary currents. It follows from Ampere s law that these boundary currents will be counteracted by currents in the coil. Hence, the computed coil current will not be realistic. For frequency domain computations the induced heating is the real part of a complex valued power. Re(z) is a non-analytical function, i.e. its derivative is not defined. Previously, COMSOL has provided a pseudo Jacobian for derivative information to the ver for the heating terms. his has caused computational problems as temperatures often were infected with small imaginary components. From version 4.1 such impurities were eliminated by removing the pseudo Jacobian contributions. he ver had then no longer information on how the coil current would influence the temperatures and the computations would not start due to one missing equation. 1 By observing that the heating term depends quadratically on the current amplitude, an elegant ution was finally found. he induced heat source can be written as: 2 (1) Qinduced I coil Qrh where Qrh is the heat source [W/m 3 ] due to a coil current of 1 A/m (as computed by COM- SOL) and I coil is the amplitude of the actual current in the coil [A/m] (to be adjusted to fulfill the constraint, c.f. chapter 2.3 below). If the uid is a metal, the magnetic field will induce high electric currents. he currents will interact with the magnetic field to create a force, the Lorentz force. his force will create a magnetic pressure that will modify the shape of the free metal surface. In addition, flow will be caused [4, 5]. he influence of a modified surface is neglected in the model, while the effect of the flow can be included by choosing a suitable heat transfer coefficient between the wall and the uid. 2.2 Heat ransfer he governing equation is: 1 According to COMSOL, AC problems can be handled by a superposition of real-valued sine and cosine terms in version 4.2a, which should ve the problem.

3 t (2) cp ( ) Qinduced where is the density [g/m 3 ], c p is the heat capacity [J/g K], is the temperature [K], is the thermal conductivity [W/m K], and Q induced is the heat source [W/m 3 ] given by equation (1). he boundary conditions are as follows: owards the coil and the additional water-cooling immediately above and below: Heat transfer coefficient At the remaining exterior boundaries: Heat transfer coefficient and radiation owards the inner uid: Heat transfer coefficient owards the inner void: Radiation owards the center line: Radial symmetry (no flux) he boundary conditions towards the inner part of the crucible are smoothed, as described in [1]. Non-linearities are included due to radiation boundary conditions and temperature dependent thermal and electrical conductivities of the crucible. 2.3 ODEs and DAEs he constraint for the current in the coil is implemented as an equation for I coil. Principally there are two alternatives: (3) W t) W ( t) otal( (4) t) ( t) Chosen ( where t is the time [s], W otal the total induced power [W], and Chosen the computed temperature in a chosen location [K], for instance the uid temperature or the temperature at a point where it can be measured. he corresponding set point values, W [W] and [K], are input to the simulation. For dynamic computations the set point values can be given as functions of the time. o find the total power COMSOL must evaluate the integral: (5) W t) 2 r Q ( r, z, t) drdz otal( induced For case studies of ing, the inner part of the container can be described by four discrete state variables: M, amount of uid [g] M, apparent amount or id (uned) material [g], temperature in the uid [K] void, average (radiation) temperature in the void [K] As simplifications it has been assumed that the temperature of the uid is uniform and that the uid and the inner surfaces facing the void exchange heat with a fictitious body of average (radiation) temperature. Depending on the process to be studied, a few more state variables can be relevant. here can, for instance, be two uid phases, or the uid(s) can contain more than one component. he equations for the state variables are: dm dt (6) minput ( t) m dm (7) m ( mtap t) dt (8) M c pl d dt W L m Q (9) Q Q 0 void_ void_w ( c pl void_ ps ( c ( Q ) 0 _w where m input is the charging rate of cold, id material [g/s], m is the effective ing rate [g/s], m is the tapping rate [g/s], tap _w cps and c pl the heat capacity of the id and the uid material [J/g K], L the heat of fusion (ing heat) [J/g], 0 the temperature of the charged id material [K], W the induced heat in the uid [W], Q void_ the heat flow from the void to the uid [W], Q the heat flow from the uid to the crucible bottom and walls [W], and Q void_w the heat flow from the void to the crucible wall and lid [W]. When cold, id material is charged, some uid will instantly freeze at the id/uid interface. At this interface, the temperature will be equal to the ing point. Due to the temperature differences there will be a heat flow from the bul uid to the id/uid ))

4 interface and from the interface to the interior of each uned particle. For a short time after a particle has been charged, the latter will be greater than the former, and uid will freeze on the particle. hen the heat flow from the bul will be the greater one, and the particle will. A comprehensive model for this process would be rather complex. We have chosen a simplified approach, where M is the apparent amount of id material in the container, as if it had stayed at the input temperature until ed. When the id material s, the model assumes that the temperature is instantly raised to the uid temperature. he ing rate is given by the following linear relation: where (10) m M ) ( is a inetic constant that can be adapted to experimental data. Furnace tapping can be included by specifying a time dependent tapping. An induction furnace can be tilted during tapping, but this effect is neglected in the model. he induced power in the uid, W, is found by integration in COMSOL. o ensure numerical stability a small, non-zero value is chosen for the electrical conductivity in the void and a gradual transition (smoothing) is applied between the high conductivity in a metal uid and the void. he height of the uid is easily computed when M and M are nown. he heat fluxes in equations (8) and (9) are given by: (11) Q R ( ) void_ void (12) Q r h ( ) ds _w 2 _w 4 4 (13) Q 2 r ( ) ds void_w w void where R is the inner radius of the crucible at the uid level, and w the effective emissivities of the uid and the wall, the Stefan-Boltzmann constant, h the heat _w transfer coefficient between the uid and the crucible, and ds the line differential along the inner boundary of the crucible and the lid. he boundary integral in equation (12) is taen along the inner crucible boundary from the center at the bottom to the uid level, while the integral in (13) is taen along the remaining part of the boundary. Both integrals are smoothed at the uid level [1]. Processes in an industrial furnace can be comparatively slow and a quasi-stationary state can be observed. For simulations, the time dependent terms in equations (2), (6), and (8) shall then be dropped. he thermal effect of changing the temperatures in the uid and the crucible will be neglected, while the power required for heating and ing the charged material is retained. he latter is a major thermal effect that can not be omitted. For COMSOL computations a stationary study is performed, without applying equation (7). he computations assume that the ing rate is equal to the charging rate, c.f. equation (6). Hence, if there is some buildup or removal of the apparent amount of id, the (presumed) ing rate must be applied for m, instead of the actual charging rate. 2.4 Solid Mechanics input For an industrial furnace the insulation needs to be soft to enable (approximately) stress-free expansion of the crucible from room temperature to operating conditions. Correspondingly, thermal expansion of the lid must be limited to prevent a significant force on the crucible. Uneven temperature distribution can cause significant thermal stresses. A stress-strain computation is therefore included, with thermal expansion as a load. he behavior is quasi-static, i.e. the temperatures change with time, while the mechanical equations are stationary. he equations, input parameters, etc., are described in COMSOL s documentation [2, 3]. When the model was tested in COMSOL Multiphysics version 4.2, we unexpectedly discovered error in the heat balance when the Solid Mechanics application mode was included. It turned out that COMSOL now would assume that the Solid Mechanics mode should deform the mesh. But it will only modify the mesh that is included in the stress-strain calculations. Hence, on the boundary between the crucible and the insulation, deformed element sizes would be applied for the crucible and undeformed ones for the insulation. Such inconsistencies will necessarily violate the heat balance. he ution is to mae sure that the mesh is not modified, i.e. the chec-box for Displacements control spatial frame must be uncheced (Solid Mechanics, Advanced Settings).

5 Figure 2 Case 3.1: emperature development Figure 4 Case 3.2: emperature development Figure 3 Case 3.1: Maximum stress in the three 3. Case Studies For the case studies we have set the frequency to 150 Hz and chosen data for artificial materials with reasonable properties. he uid in the crucible is initially at 850 C, 50 C above its ing point. hen 1000 g bul id shall be charged at a constant rate during 4 hours, where after the uid temperature shall be raised to 1000 C. Smooth changes in dynamic input data was chosen to ensure numerical stability. he input feed rate was raised from zero to a constant value during 30 s at the start and then lowered for 5 min after 4 hours. he longer period at the end was chosen as 2.5 times the maximum time step. Higher accuracy was demanded for the non-linear iterations than for the time stepping, c.f. recommendations in [1]. 3.1 Non-conductive Liquid, Power Control Power control applies equation (3) to compute the electric current in the coil. First a proper stationary state was computed. hen the power set point was raised during 30 s to a level sufficient to the input material and heat it to 850 C. Heat calculations showed that the uid temperature would be raised to Figure 5 Case 3.2: Maximum stress in the three 1000 C if the power level was ept 31 min longer than the charging period. But it is required to heat the crucible and the insulation at the same time and the high power was maintained for another 59 min (after a few trial runs). Some computed temperatures and maximum stresses are shown in Figure 2 and Figure Non-conductive Liquid, emperature Control Now equation (4) was ved with as Chosen. First a stationary state was computed, followed by a dynamic computation. Some test simulations revealed that temperature control according to equation (4) is rather stiff and more smooth transitions were chosen. he charging rate was now raised during 120 s. Reducing the charging rate after some 4 hours was difficult. For the first tests the computed power would drop to zero and the computations would then stop due to an error condition. After some trial and error one hour diminishing period, partly overlapped by the temperature increase to 1000 C, wored reasonably well. Some computed temperatures and maximum stresses are shown in Figure 4 and Figure 5.

6 Figure 6 Case 3.3: emperature development Figure 8 Case 3.4: emperature development Figure 7 Case 3.3: Maximum stress in the three 3.3 Non-conductive Liquid, Combined Control A combined control was also tested. Formally power control was applied, using equation (3). hen proportional temperature control was included for additional power adjustment: (14) W W ) _0 ( where W _0 is the power set point that was applied for case 3.1, was set to 850 C during the first 4 hours and then raised smoothly to 1000 C during 1½ hours. is a constant for tuning. For a very low value of, the last term in the equation is negligible and the combined controller will essentially wor as pure power control. For a high value of the parameter, the uid temperature will be close to its set point and the controller will be close to pure temperature control. 2 [W/K] was chosen for our case study. Some computed temperatures and maximum stresses are shown in Figure 6 and Figure Conductive Liquid, Combined Control Combined control was chosen for a test case with a conducting uid. he electric conduc- Figure 9 Case 3.4: Maximum stress in the three tivity of the uid was raised to a level typical for metals, while the crucible was treated as an electric insulator. he remaining input was the same as in the previous case. Some computed temperatures and maximum stresses are shown in Figure 8 and Figure 9. As the power is now induced in the uid, temperature gradients and thermal stresses in the crucible are considerably reduced compared to case Quasi-stationary computations Quasi-stationary computations have been run and compared with corresponding dynamic simulations. For stable periods the quasistationary results agree very well with the corresponding dynamic ones. 3.6 Other Experience and Comments he model was first converted from COM- SOL Multiphysics version 3.5a to version 4.0a. While the Matlab interface was required for the computational procedure in version 3.5a, it was now possible to set up proper studies in the new COMSOL GUI. o perform a stationary study the magnetic equations are ved first. his ution is then applied as initial values for the magnetic field for a computation of the complete system.

7 he conversion revealed issues that did not wor properly in the new COMSOL version. he faults have been corrected one by one by COMSOL and the model now performs well for COMSOL Multiphysics version 4.2. Dynamic input is now easily described by functions. he functions can depend on various parameters and user functions can be applied in the description of other functions. While our previous model applied Matlab to compute discrete changes, these are now implemented as rapid, dynamic changes. he user effort is far less than before, and the simulations more realistic. he computations are now faster. While the previous model applied some 12 iterations for a typical stationary case, only around 7 is required in version 4.2. For the time integration we have applied the experience from the previous testing [1]: Very low relative tolerance: Individual, suitable abute tolerances for each group of variables Maximum order 2 for the time stepping Not to high maximum time step (2 min) (Non-linear) tolerance factor: 0.1 or less For case 3.4 we found large oscillations in the computed coil current and the power during part of the simulation. he oscillations disappeared after the non-linear tolerance factor had been reduced (from 0.1 to 0.001), in accordance with our previous experience [1]. 4. Discussion he quasi-stationary option is a valuable tool for studying the effect of design issues lie geometry (crucible, insulation, location of the coil, etc.), material properties, and frequency. Alternative designs can be studied for typical operational conditions and ey figures for each design can easily be compared. Dynamic simulations are far more time consuming (man hours and computer time) and can be more difficult to compare. Dynamic simulations are required to study operating procedures, especially to get improved insight in transient behavior. Our case studies show that flexible options for control can be important. For the previous version, only power control was available, but it is difficult to manually adjust the power to obtain a requested temperature evolution. he final temperature in Figure 2 is for instance lower than the target value of 1000 C. emperature control ensures correct target temperature, but can involve unrealistic, even unphysical, power levels. If a rapid temperature increase is specified, the required power can be extremely high, while a too fast power decrease would need negative power. Combined power and temperature control is a suitable option than can be made even more flexible with a time dependent tuning factor,. More sophisticated control, lie a PID controller, can easily be included. 5. Conclusions A multiphysics model for industrial induction furnaces has successfully been converted from COMSOL version 3.5a, via versions 4.0a and 4.1, to version 4.2. he model has been extended to handle: Quasi-stationary cases Liquid that is either conductive or nonconductive Far more complex dynamic cases Either the total induced power, a temperature, or a combination can be specified (to determine the coil current). he model can be applied to study operational conditions, thermal stresses, or design details for a high temperature reactor. 6. References 1. S. A. Halvorsen, A Model for High emperature Inductive Heating, paper presented at the COMSOL Conference, Milano, Italy (2009) 2. COMSOL Multiphysics User s Guide, Version 4.2 (2011) 3. COMSOL Multiphysics, Structural Mechanics Module, User s Guide, Version 4.2 (2011). 4. P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press (2001) 5. N. Kleinnecht and S. A. Halvorsen, High Frequency Magnetohydrodynamic Calculations in COMSOL Multiphysics, poster presented at the COMSOL Conference, Stuttgart, Germany (2011) 7. Acnowledgements he authors would lie to than Aust- and Vest-Agder counties and the Competence Development Fund of Southern Norway for their financial support.